Average Error: 34.9 → 9.1
Time: 17.7s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.773941363092559658836993715800348666263 \cdot 10^{46}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1.052552064964455729256789034532554564117 \cdot 10^{-131}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.711794221765945466011689717230842352302 \cdot 10^{108}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -2.773941363092559658836993715800348666263 \cdot 10^{46}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -1.052552064964455729256789034532554564117 \cdot 10^{-131}:\\
\;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2 \cdot a}\\

\mathbf{elif}\;b \le 1.711794221765945466011689717230842352302 \cdot 10^{108}:\\
\;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r59566 = b;
        double r59567 = -r59566;
        double r59568 = r59566 * r59566;
        double r59569 = 4.0;
        double r59570 = a;
        double r59571 = c;
        double r59572 = r59570 * r59571;
        double r59573 = r59569 * r59572;
        double r59574 = r59568 - r59573;
        double r59575 = sqrt(r59574);
        double r59576 = r59567 - r59575;
        double r59577 = 2.0;
        double r59578 = r59577 * r59570;
        double r59579 = r59576 / r59578;
        return r59579;
}
↓
double f(double a, double b, double c) {
        double r59580 = b;
        double r59581 = -2.7739413630925597e+46;
        bool r59582 = r59580 <= r59581;
        double r59583 = -1.0;
        double r59584 = c;
        double r59585 = r59584 / r59580;
        double r59586 = r59583 * r59585;
        double r59587 = -1.0525520649644557e-131;
        bool r59588 = r59580 <= r59587;
        double r59589 = 4.0;
        double r59590 = a;
        double r59591 = r59590 * r59584;
        double r59592 = r59589 * r59591;
        double r59593 = r59580 * r59580;
        double r59594 = r59593 - r59592;
        double r59595 = sqrt(r59594);
        double r59596 = r59595 - r59580;
        double r59597 = r59592 / r59596;
        double r59598 = 2.0;
        double r59599 = r59598 * r59590;
        double r59600 = r59597 / r59599;
        double r59601 = 1.7117942217659455e+108;
        bool r59602 = r59580 <= r59601;
        double r59603 = -r59580;
        double r59604 = r59603 - r59595;
        double r59605 = 1.0;
        double r59606 = r59605 / r59599;
        double r59607 = r59604 * r59606;
        double r59608 = 1.0;
        double r59609 = r59580 / r59590;
        double r59610 = r59585 - r59609;
        double r59611 = r59608 * r59610;
        double r59612 = r59602 ? r59607 : r59611;
        double r59613 = r59588 ? r59600 : r59612;
        double r59614 = r59582 ? r59586 : r59613;
        return r59614;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original34.9
Target21.4
Herbie9.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -2.7739413630925597e+46

    1. Initial program 58.0

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    2. Taylor expanded around -inf 4.1

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]

    if -2.7739413630925597e+46 < b < -1.0525520649644557e-131

    1. Initial program 38.9

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
    2. Using strategy rm

    3. Applied flip--38.9

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]

    4. Simplified17.4

      \[\leadsto \frac{\frac{\color{blue}{0 + \left(a \cdot c\right) \cdot 4}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]

    5. Simplified17.4

      \[\leadsto \frac{\frac{0 + \left(a \cdot c\right) \cdot 4}{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}{2 \cdot a}\]

    if -1.0525520649644557e-131 < b < 1.7117942217659455e+108

    1. Initial program 11.7

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
    2. Using strategy rm

    3. Applied div-inv11.9

      \[\leadsto \color{blue}{\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}}\]

    if 1.7117942217659455e+108 < b

    1. Initial program 49.7

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    2. Taylor expanded around inf 3.0

      \[\leadsto \color{blue}{1 \cdot \frac{c}{b} - 1 \cdot \frac{b}{a}}\]

    3. Simplified3.0

      \[\leadsto \color{blue}{1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.773941363092559658836993715800348666263 \cdot 10^{46}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1.052552064964455729256789034532554564117 \cdot 10^{-131}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.711794221765945466011689717230842352302 \cdot 10^{108}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))